Anti-eikonal equation of an eigenmirror.

We call a surface that appears undistorted when viewed in a curved mirror an eigensurface and the mirror an eigenmirror. Such pairs are described by a first-order nonlinear partial differential equation of the form a0+a1ux+a2uy+a3uxuy+a4ux2+a5uy2=0, where ai=ai(x,y,u), which we call the anti-eikonal equation. We give examples of symbolic and numerical solutions, including pairs that are geometrically congruent. Ray tracing simulations are included that visually confirm the unusual properties of these surfaces.

Solution to the bundle-to-bundle mapping problem of geometric optics using four freeform reflectors.

Here we present a method for the coupled design of four freeform reflective surfaces that will control a bundle of rays. By this, we mean that given an input bundle of rays, we can construct an optical system that will map it to a given output bundle, where a ray-to-ray correspondence is realized as per the prescribed data. The method makes use of the Cartan-Kähler theorem of exterior differential systems. Sample imaging applications are given.

Designing coupled free-form surfaces.

The problem of designing optical systems that contain free-form surfaces is a challenging one, even in the case of designing a single surface. Here we present a method for the coupled design of two free-form reflective surfaces that will have a prescribed distortion. On one hand, the method can be described using traditional vectors and matrices, which we do, but it is motivated by viewing the problem in the language of distributions from differential geometry and makes use of the exterior differential systems, which we relegate to an appendix. Example applications are given to the design of a mirror pair that increases the field of view of an observer, a similar mirror pair that also rotates the observer's view, and a pair of mirrors that give the observer a traditional panoramic strip view of the scene.

Controlling a ray bundle with a free-form reflector.

The problem of controlling a single ray bundle with a single reflector is not generally solvable, but approximate solutions may often be found that are acceptable for applications. We introduce a new technique for finding such approximations and apply it to the design of a driver-side mirror for an automobile that has no blind spot and minimal distortion.

Programmable imaging with two-axis micromirrors.

We demonstrate a means of creating a digital image by using a two-axis tilt micromirror to scan a scene. For each different orientation we extract a single gray scale value from the mirror and combine them to form a single composite image. This allows one to choose the distribution of the samples, and so in principle a variable resolution image could be created. We demonstrate this ability to control resolution and projection by constructing a voltage table that compensates for the nonlinear response of the mirrors to the applied voltage.

Realizing any central projection with a mirror pair.

We show that, for any rotationally symmetric projection with a single virtual viewpoint, it is possible to design a two-mirror rotationally symmetric system that realizes the projection exactly. These mirror pairs are derived from two coupled differential equations. We give examples in which the projections from the sphere at infinity are stereographic, perspective, and equiresolution.

Designing a mirror to realize a given projection.

I present a design technique for realizing given projections as catadioptric sensors. In general, these problems do not have solutions, but approximate solutions may often be found that are visually acceptable. The method described reduces the problem to solving a linear system. A given transformation from the image plane to an object surface is shown to determine a vector field that is normal to the surface in the case where the vector field is a gradient. For the case when the vector field is not a gradient, several functionals are presented that may be minimized to give approximate solutions. As an application several new designs are described, including a mirror that directly gives a full 360-deg cylindrical projection without the need for any digital processing.

Blind-spot problem for motor vehicles.

We consider the problem of designing a passenger side mirror for an automobile that does not have a blind spot and that does not distort the image. Our model consists of a coupled pair of partial differential equations that do not have a common solution. Using a best mean-square-error functional, we find approximate solutions using nonlinear optimization. In one case a local minimum provides a mirror that solves the problem, but it does not reverse the image.